Particle accelerator (Homework 8-323)

1. In an accelerator, electrons and positrons collide head-on with each other at an interaction point. The particles have speed 0.6 c.  Suppose the annihilation products consist solely of photons. a. Show there must be two or more photons in the final state. b. In the case where two photons are emitted, show that the photons are back-to-back with equal energies (in the lab frame). c. In the c
ase where two photons are emitted, find the energy of each photon, find the wavelength of the photons (in the lab frame). 

a)
Suppose the reaction would have been
$e^++e^-=γ$
Then in the system of mass center the initial total momentum is zero. The momentum of the resulting gamma (photon) is
$p_γ=E_γ/c$
And therefore the total momentum conservation is not respected. Therefore there results at least two photons γ for the total momentum (in the SCM) to be conserved:
$e^++e^-=γ+γ$
Being emitted in opposite directions each with a momentum and an energy (in SCM)
$|p_γ |=Eγ/c$    ;   $E_γ+E_γ=2mc^2$    
where m is the mass of electron (positron)in SCM    $m=m_0/\sqrt{1-β^2}$
The energy of the photons is the same in the SCM and laboratory frame.
$E_γ=mc^2=(m_0 c^2)/\sqrt{1-β^2}=(9.1*10^{-31}*9*10^{16})/\sqrt{1-0.6^2}=$
$=1.024*10^{-13}  J=0.64 MeV$
The wavelength of each photon is
$E_γ=hc/λ$     or
$λ=hc/E_γ =(6.626*10^{-34}*3*10^8)/(1.024*10^{-13} )=1.94*10^{-12}  m=1.94 pm$